Geometry and Topology of Holomorphic Symplectic Varieties

代数几何是一门研究代数簇的学科，代数簇是由多项式方程的解描述的空间。这门学科始于古希腊人关于椭圆、抛物线和双曲线的工作，现在它已经发展成为一门现代学科，与物理学和计算机科学有着许多丰富的联系。由这个NSF奖支持的研究将集中在一类重要的代数簇，称为全纯辛簇，位于几何，表示论和数学物理的十字路口。在这个项目中，PI将建立紧和非紧全纯辛簇之间的联系，重点是P=W猜想，这是连接Hitchin可积系统和Hodge理论的中心猜想。紧全纯辛簇是K3曲面的高维推广，几十年来一直受到人们的广泛研究。一个系统的方法来比较紧凑和非紧凑的几何形状，并引入新的技术，希钦系统从紧凑的全纯辛品种将带来新的观点和富有成效的成果，P=W猜想和相关的questions.This奖项反映了NSF的法定使命，并已被认为是值得支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

Algebraic geometry is a subject studying algebraic varieties, which are spaces described by solutions of polynomial equations. This subject started with the work of the ancient Greeks about ellipses, parabolas, and hyperbolas, and now it has grown to be a modern subject sharing many rich connections with physics and computer science. The research supported by this NSF award will focus on an important class of algebraic varieties, called the holomorphic symplectic varieties, that lie at the crossroads of geometry, representation theory, and mathematical physics. A better understanding of these varieties will bring deep insights in both algebraic geometry and its related fields.In this project, the PI will build connections between compact and noncompact holomorphic symplectic varieties with a focus on the P=W conjecture, which is a central conjecture connecting Hitchin's integrable systems and Hodge theory. Compact holomorphic symplectic varieties, which are higher dimensional generalizations of K3 surfaces, have been studied intensively for decades. A systematic way to compare compact and noncompact geometries, and to introduce new techniques to Hitchin systems from compact holomorphic symplectic varieties will bring fresh perspectives and fruitful outcomes to the P=W conjecture and related questions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.